Equivalence & Bi-implication

The Converse of Statement:

Let A → B be an implication, then the converse of A → B is  B → A

Equivalent Statements:

Let the implication X → Y be true, if Y → X is true, then X and Y are said to be equivalent statements. Thus, we write and say that X ↔ Y is a bi-implication.

Example 7.6.1:

Given the statements:

P₁: The teacher entered and started the lesson

P₂: Either it will rain tomorrow or the visit will take place, or it will rain tomorrow and the visit will take place anyway.

Use appropriate symbols and truth tables to describe:

1. The conditions for P₁ to be true,
2. The conditions under which P₂ will be false

Solution:

1. Let the sub-statements be

A: the teacher entered

B: the teacher started the lesson

If P₁ is true, then A → B is true.

Recall, the relation “and” which suggests both A and B must true before A → B could be true

Truth table